PACKING CIRCLES WITHIN CIRCULAR CONTAINERS: A NEW HEURISTIC ALGORITHM FOR THE BALANCE CONSTRAINTS CASE

Oliveira, Washington Alves de; Salles Neto, Luiz Leduino de; Moretti, Antonio Carlos; Reis, Ednei Felix

doi:10.1590/0101-7438.2016.036.02.0279

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Articles • Pesqui. Oper. 36 (2) • May-Aug 2016 • https://doi.org/10.1590/0101-7438.2016.036.02.0279 copy

PACKING CIRCLES WITHIN CIRCULAR CONTAINERS: A NEW HEURISTIC ALGORITHM FOR THE BALANCE CONSTRAINTS CASE

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In this work we propose a heuristic algorithm for the layout optimization for disks installed in a rotating circular container. This is a unequal circle packing problem with additional balance constraints. It proved to be an NP-hard problem, which justifies heuristics methods for its resolution in larger instances. The main feature of our heuristic is based on the selection of the next circle to be placed inside the container according to the position of the system's center of mass. Our approach has been tested on a series of instances up to 55 circles and compared with the literature. Computational results show good performance in terms of solution quality and computational time for the proposed algorithm.

Keywords:
packing problem; layout optimization problem; nonidentical circles; heuristic algorithm

Size	Radii	Mass		Size	Radii	Mass
5	[20.71, 50]	[20.71, 50]		7	[8.5, 12]	[72.25, 144]
9	[0.17, 0.41]	[0.17, 0.41]		40	[81, 120]	[6, 14]
5	[6, 11]	[36, 121]
			Second set of instances (^{Huang & Chen, 2006}3 HUANG W & CHEN M. 2006. Note on: An improved algorithm for the packing of unequal circles within a larger containing circle. Computers & Industrial Engineering, 50(3): 338-344.)
Size	Radii	Mass		Size	Radii	Mass
7	[20, 20]	[400, 400]		17	[5, 25]	[25, 625]
12	[23.72, 100]	[562.64, 10⁴]		37	[20, 20]	[400, 400]
15	[1, 15]	[1, 225]		50	[20, 20]	[400, 400]
			Third of instances (^{Xiao et al., 2007}15 XIAO RB, XU YC & AMOS M. 2007. Two hybrid compaction algorithms for the layout optimization problem. Biosystems, 90(2): 560-567.)
Size	Radii	Mass		Size	Radii	Mass
10	[5, 23]	[20, 93]		35	[7, 24]	[10, 99]
15	[6, 24]	[12, 98]		40	[6, 23]	[12, 99]
20	[5, 24]	[11, 94]		45	[6, 24]	[11, 99]
25	[6, 24]	[11, 96]		50	[5, 24]	[10, 99]
30	[6, 24]	[12, 97]		55	[6, 24]	[13, 99]

Instance	n	(^{Huang & Chen, 2006}3 HUANG W & CHEN M. 2006. Note on: An improved algorithm for the packing of unequal circles within a larger containing circle. Computers & Industrial Engineering, 50(3): 338-344.)			(^{Lei, 2009}4 LEI K. 2009. Constrained layout optimization based on adaptive particle swarm optimizer. Springer-Verlag Berlin Heidelberg, 1: 434-442.)			(^{Liu & Li, 2010}9 LIU J, LI G, CHEN D, LIU W & WANG Y. 2010. Two-dimensional equilibrium constraint layout using simulated annealing. Computers & Industrial Engineering, 59(4): 530-536.)
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
1.1	5	120.710	7.63 × 10^-4	3.82	-	-	-	120.710678	3.89 × 10^-9	0.594
1.2	9	-	-	-	-	-	-	1.000000	9.7 × 10^-9	2.891
1.3	5	22.327	3.90 × 10^-4	2.91	-	-	-	22.249000	3.12 × 10^-9	1.313
1.4	7	31.981	4.11 × 10^-3	76.33	31.924	1.40 × 10^-5	427	31.854600	1.5 × 10^-9	54.828
1.5	40	742.750	5.40 × 10^-4	12.27	769.819	3.25 × 10^-4	1724	725.043500	2.47 × 10^-9	7.187
Instance	n	(^{Liu et al., 2010}9 LIU J, LI G, CHEN D, LIU W & WANG Y. 2010. Two-dimensional equilibrium constraint layout using simulated annealing. Computers & Industrial Engineering, 59(4): 530-536.)			(^{Liu et al., 2011}10 LIU J, LI G & GENG H. 2011. A new heuristic algorithm for the circular packing problem with equilibrium constraints. Science China Information Sciences, 54(8): 1572-1584.)
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
1.1	5	120.7106782	4.30 × 10^-9	0.696	120.710678	3.25 × 10^-9	0.350
1.2	9	-	-	-	1.000000	2.38 × 10^-9	0.225
1.3	5	22.2463	3.52 × 10^-9	0.469	22.246203	2.63 × 10^-9	0.506
1.4	7	31.8412	2.03 × 10^-10	58.797	31.841134	5.89 × 10^-9	27.766
1.5	40	719.6782	2.86 × 10^-9	154.594	716.610932	1.77 × 10^-9	63.263
Instance	n	(^{He et al., 2013}2 HE K, MO D, YE T & HUANG W. 2013. A coarse-to-fine quasi-physical optimization method for solving the circle packing problem with equilibrium constraints. Computers & Industrial Engineering, 66(4): 1049-1060.)			(^{Liu et al., 2015}7 LIU J, JIANG Y, LI G, XUE Y, LIU Z & ZHANG Z. 2015. Heuristic-based energy landscape paving for the circular packing problem with performance constraints of equilibrium. Physica A: Statistical Mechanics and its Applications, 431: 166-174.)			Our algorithm
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
1.1	5	120.710678	2.11 × 10^-10	0.046	120.710678	3.49 × 10^-10	-	120.710618	0	1.925
1.2	9	1.000000	2.00 × 10^-12	0.172	1.000000	9.92 × 10^-10	-	1.000000	0	2.365
1.3	5	22.246203	3.44 × 10^-10	0.15	22.246203	9.22 × 10^-10	-	23.624138	0	1.812
1.4	7	31.841134	1.80 × 10^-10	98.031	31.841133	1.13 × 10^-9	-	31.905109	0	110.912
1.5	40	709.812500	1.85 × 10^-10	434.734	715.978266	3.15 × 10^-7	-	745.118349	0	829.156

Instance	n	(^{Huang & Chen, 2006}3 HUANG W & CHEN M. 2006. Note on: An improved algorithm for the packing of unequal circles within a larger containing circle. Computers & Industrial Engineering, 50(3): 338-344.)			(^{Liu et al., 2010}9 LIU J, LI G, CHEN D, LIU W & WANG Y. 2010. Two-dimensional equilibrium constraint layout using simulated annealing. Computers & Industrial Engineering, 59(4): 530-536.)			(^{Liu, 2011}10 LIU J, LI G & GENG H. 2011. A new heuristic algorithm for the circular packing problem with equilibrium constraints. Science China Information Sciences, 54(8): 1572-1584.)
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
2.1	7	60.00	3.6 × 10^-3	7.78	60.0000	1.59 × 10^-9	1.031	60.000000	7.38 × 10^-9	0.206
2.2	12	215.470	9.5 × 10^-3	44.78	215.470054	2.74 × 10^-7	11.562	215.470054	3.77 × 10^-7	3.042
2.3	15	39.780	7.6 × 10^-3	91.92	39.123400	2.97 × 10^-9	130.362	39.065105	4.13 × 10^-9	365.797
2.4	17	49.720	5.1 × 10^-3	157.92	49.507800	6.83 × 10^-8	123.407	49.368194	2.50 × 10^-8	597.719
2.5	37	135.176	6.7 × 10^-3	18.29	135.175410	1.23 × 10^-8	0.203	135.175410	4.91 × 10^-8	0.426
2.6	50	159.570	8.0 × 10^-3	348.97	158.967800	4.81 × 10^-9	453.625	158.967698	3.49 × 10^-9	1078.386
Instance	n	(^{He et al., 2013}2 HE K, MO D, YE T & HUANG W. 2013. A coarse-to-fine quasi-physical optimization method for solving the circle packing problem with equilibrium constraints. Computers & Industrial Engineering, 66(4): 1049-1060.)			(^{Liu et al., 2015}7 LIU J, JIANG Y, LI G, XUE Y, LIU Z & ZHANG Z. 2015. Heuristic-based energy landscape paving for the circular packing problem with performance constraints of equilibrium. Physica A: Statistical Mechanics and its Applications, 431: 166-174.)			Our algorithm
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
2.1	7	60.000000	2.66 × 10^-9	0.015	60.000000	5.13 × 10^-9	-	60.000000	0	0.118
2.2	12	215.470054	2.92 × 10^-9	0.391	215.470054	1.69 × 10^-7	-	215.470054	0	42.125
2.3	15	39.037635	7.78 × 10^-10	36.343	39.065105	1.12 × 10^-8	-	40.090538	0	167.248
2.4	17	49.338750	1.51 × 10^-9	10.562	49.368194	1.42 × 10^-8	-	50.246873	0	325.080
2.5	37	135.175410	5.84 × 10^-9	0.234	135.175410	3.35 × 10^-9	-	145.624231	0	0.402
2.6	50	158.963672	5.14 × 10^-9	268.078	158.967698	3.65 × 10^-9	-	163.206698	0	0.309

Instance	n	(^{Xiao et al., 2007}15 XIAO RB, XU YC & AMOS M. 2007. Two hybrid compaction algorithms for the layout optimization problem. Biosystems, 90(2): 560-567.) (SA)			(^{Xiao et al., 2007}15 XIAO RB, XU YC & AMOS M. 2007. Two hybrid compaction algorithms for the layout optimization problem. Biosystems, 90(2): 560-567.) (PSO)
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
3.1	10	69.96	0	3237	59.93	0	2898
3.2	15	68.77	0	8320	67.65	0	8659
3.3	20	83.09	0	18431	83.06	0	20035
3.4	25	83.97	0	34032	84.24	0	36815
3.5	30	99.58	0	54565	99.89	0	62360
3.6	35	102.86	0	76760	102.71	0	86537
3.7	40	115.15	0	128112	115.58	0	122390
3.8	45	120.63	0	1674848	119.67	0	153006
3.9	50	125.82	0	198071	126.19	0	199050
3.10	55	138.22	0	198071	138.89	0	244171
Instance	n	(^{Liu et al., 2015}7 LIU J, JIANG Y, LI G, XUE Y, LIU Z & ZHANG Z. 2015. Heuristic-based energy landscape paving for the circular packing problem with performance constraints of equilibrium. Physica A: Statistical Mechanics and its Applications, 431: 166-174.)			Our algorithm
		f ₁	f ₂	time(s)	f ₁	f ₂	time(s)
3.1	10	59.92	5.40 × 10^-9	-	59.96	0	28
3.2	15	67.39	6.44 × 10^-9	-	67.59	0	39
3.3	20	82.99	2.48 × 10^-9	-	83.85	0	82
3.4	25	82.98	5.10 × 10^-9	-	83.68	0	107
3.5	30	98.97	4.14 × 10^-9	-	98.95	0	134
3.6	35	102.32	4.50 × 10^-9	-	105.15	0	155
3.7	40	115.00	5.82 × 10^-9	-	114.15	0	96
3.8	45	119.07	7.64 × 10^-9	-	117.69	0	118
3.9	50	124.98	4.45 × 10^-9	-	124.32	0	136
3.10	55	135.13	2.19 × 10^-9	-	137.37	0	154

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